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A285321 Square array A(1,k) = A019565(k), A(n,k) = A065642(A(n-1,k)), read by descending antidiagonals. 10

%I #24 Mar 27 2021 06:17:18

%S 2,3,4,6,9,8,5,12,27,16,10,25,18,81,32,15,20,125,24,243,64,30,45,40,

%T 625,36,729,128,7,60,75,50,3125,48,2187,256,14,49,90,135,80,15625,54,

%U 6561,512,21,28,343,120,225,100,78125,72,19683,1024

%N Square array A(1,k) = A019565(k), A(n,k) = A065642(A(n-1,k)), read by descending antidiagonals.

%C A permutation of the natural numbers > 1.

%C Otherwise like array A284311, but the columns come in different order.

%H Antti Karttunen, <a href="/A285321/b285321.txt">Table of n, a(n) for n = 1..120; the first 15 antidiagonals of array</a>

%F A(1,k) = A019565(k), A(n,k) = A065642(A(n-1,k)).

%F For all n >= 2: A(A008479(n), A087207(n)) = n.

%e The top left 12x6 corner of the array:

%e 2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 42, 35

%e 4, 9, 12, 25, 20, 45, 60, 49, 28, 63, 84, 175

%e 8, 27, 18, 125, 40, 75, 90, 343, 56, 147, 126, 245

%e 16, 81, 24, 625, 50, 135, 120, 2401, 98, 189, 168, 875

%e 32, 243, 36, 3125, 80, 225, 150, 16807, 112, 441, 252, 1225

%e 64, 729, 48, 15625, 100, 375, 180, 117649, 196, 567, 294, 1715

%t a065642[n_] := Module[{k}, If[n == 1, Return[1], k = n + 1; While[ EulerPhi[k]/k != EulerPhi[n]/n, k++]]; k];

%t A[1, k_] := Times @@ Prime[Flatten[Position[#, 1]]]&[Reverse[ IntegerDigits[k, 2]]];

%t A[n_ /; n > 1, k_] := A[n, k] = a065642[A[n - 1, k]];

%t Table[A[n - k + 1, k], {n, 1, 10}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Nov 17 2019 *)

%o (Scheme)

%o (define (A285321 n) (A285321bi (A002260 n) (A004736 n)))

%o (define (A285321bi row col) (if (= 1 row) (A019565 col) (A065642 (A285321bi (- row 1) col))))

%o (Python)

%o from operator import mul

%o from sympy import prime, primefactors

%o def a019565(n): return reduce(mul, (prime(i+1) for i, v in enumerate(bin(n)[:1:-1]) if v == '1')) if n > 0 else 1 # This function from _Chai Wah Wu_

%o def a007947(n): return 1 if n<2 else reduce(mul, primefactors(n))

%o def a065642(n):

%o if n==1: return 1

%o r=a007947(n)

%o n = n + r

%o while a007947(n)!=r:

%o n+=r

%o return n

%o def A(n, k): return a019565(k) if n==1 else a065642(A(n - 1, k))

%o for n in range(1, 11): print([A(k, n - k + 1) for k in range(1, n + 1)]) # _Indranil Ghosh_, Apr 18 2017

%Y Transpose: A285322.

%Y Cf. A019565, A065642.

%Y Cf. A008479 (index of the row where n is located), A087207 (of the column).

%Y Cf. arrays A284311, A285325, also A285332.

%K nonn,tabl

%O 1,1

%A _Antti Karttunen_, Apr 17 2017

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Last modified March 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)